1 Purpose

2 Specification

3 Description

S17ACF evaluates an approximation to the Bessel function of the second kind Y0x.

Note:Y0x is undefined for x≤0 and the routine will fail for such arguments.

The routine is based on four Chebyshev expansions:

For 0<x≤8,

Y0x=2πln⁡x∑r=0′arTrt+∑r=0′brTrt, with ​t=2x82-1.

For x>8,

Y0x=2πxP0xsinx-π4+Q0xcosx-π4

where P0x=∑′r=0crTrt,

and Q0x=8x∑′r=0drTrt,with ​t=28x2-1.

For x near zero, Y0x≃2πlnx2+γ, where γ denotes Euler's constant. This approximation is used when x is sufficiently small for the result to be correct to machine precision.

For very large x, it becomes impossible to provide results with any reasonable accuracy (see Section 7), hence the routine fails. Such arguments contain insufficient information to determine the phase of oscillation of Y0x; only the amplitude, 2πn, can be determined and this is returned on soft failure. The range for which this occurs is roughly related to machine precision; the routine will fail if x≳1/machine precision (see the Users' Note for your implementation for details).

5 Parameters

On entry: IFAIL must be set to 0, -1​ or ​1. If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction for details.

For environments where it might be inappropriate to halt program execution when an error is detected, the value -1​ or ​1 is recommended. If the output of error messages is undesirable, then the value 1 is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is 0. When the value -1​ or ​1 is used it is essential to test the value of IFAIL on exit.

On exit: IFAIL=0 unless the routine detects an error or a warning has been flagged (see Section 6).

6 Error Indicators and Warnings

If on entry IFAIL=0 or -1, explanatory error messages are output on the current error message unit (as defined by X04AAF).

Errors or warnings detected by the routine:

IFAIL=1

X is too large. On soft failure the routine returns the amplitude of the Y0 oscillation, 2/πx.

IFAIL=2

X≤0.0, Y0 is undefined. On soft failure the routine returns zero.

7 Accuracy

Let δ be the relative error in the argument and E be the absolute error in the result. (Since Y0x oscillates about zero, absolute error and not relative error is significant, except for very small x.)

If δ is somewhat larger than the machine representation error (e.g., if δ is due to data errors etc.), then E and δ are approximately related by

E≃xY1xδ

(provided E is also within machine bounds). Figure 1 displays the behaviour of the amplification factor xY1x.

However, if δ is of the same order as the machine representation errors, then rounding errors could make E slightly larger than the above relation predicts.

For very small x, the errors are essentially independent of δ and the routine should provide relative accuracy bounded by the machine precision.

For very large x, the above relation ceases to apply. In this region, Y0x≃2πxsinx-π4. The amplitude 2πx can be calculated with reasonable accuracy for all x, but sinx-π4 cannot. If x-π4 is written as 2Nπ+θ where N is an integer and 0≤θ<2π, then sinx-π4 is determined by θ only. If x≳δ-1, θ cannot be determined with any accuracy at all. Thus if x is greater than, or of the order of the inverse of machine precision, it is impossible to calculate the phase of Y0x and the routine must fail.

Figure 1

8 Further Comments

None.

9 Example

This example reads values of the argument x from a file, evaluates the function at each value of x and prints the results.